Mechanics of 21st Century - ICTAM04 Proceedings
MODELLING SURFACE TENSION USING A GHOST FLUID TECHNIQUE
WITHIN A VOLUME OF FLUID FORMULATION
Marianne M. Francois , Douglas B. Kothe, Sharen J. Cummins
Computer and Computational Sciences Division, Los Alamos National Laboratory,
Los Alamos, NM 87545, U.S.A.
Summary We investigate a ghost fluid method in a volume of fluid formulation to model the surface tension force along
the normal direction at an interface between two fluids of different properties. We compare it with the Continuum Surface
Tension Force (CSF) method using numerical results to support our findings.
MODELING SURFACE TENSION
The continuum surface tension force (CSF) of Brackbill et al. [1] has been widely used over the past twelve years to
model surface tension in multiphase flow in volume of fluid (VOF), level-set (LS) and front tracking (FT) methods. Surface
tension forces acting on the interface are transformed to volume forces in regions near to the interface via delta
functions, leading to ideally discontinuous interface jump conditions being modelled as smooth. Recently, ghost fluid
methods (GFM) were presented in [3, 4] to impose ‘sharper’ boundary conditions on embedded boundaries. GFM have
been employed to model surface tension in conjunction with LS techniques, since GFM require knowledge of the
distance from the interface, which is the information natural for LS methods. We denote VOF-GFM as an underlying VOF
method that employs GFM for surface tension, VOF-CSF as an underlying VOF method that employs the CSF method for
surface tension, and LS -GFM as an underlying LS approach that employs GFM to model surface tension. VOF-GFM is an
interesting alternative to LS-GFM because of the superiority of VOF methods in mass conservation over LS methods.
However, since in VOF methods volume fractions are employed to track the interface, a distance function is not naturally
available. We have therefore devised a novel technique [5] to reconstruct distance functions from volume fractions,
allowing one to combine GFM with VOF. We now address whether or not there are advantages in using a VOF-GFM to
model surface tension flows over the widely-used, classical VOF-CSF method. To aid our comparison, we consider the
case of a static drop in equilibrium. The jump condition in pressure along the normal direction is [P]=σκ, where σ is the
surface tension coefficient and κ is the interfacial curvature. In VOF methods, a discountinous Heaviside function
(volume fractions) f on a fixed grid is employed to track the interface. In cells filled with fluid 1, f is equal to 1, and in cells
filled with fluid 2, f is equal to 0. For cells containing the interface, f is between 0 and 1. In VOF, a single set of governing
equations are considered with an additional evolution equation to track the interface. In our case, the VOF is initialized
using a recursive local mesh refinement technique for cells that contains the interface. We consider the fluid inviscid and
the flow incompressible. The governing equations
are:
r
r
∇u
. =0
(1) ,
∂( ρu )
rr
+ ∇.( ρuu ) = −∇P
∂t
(2) ,
∂f r
+ u.∇f = 0
∂t
(3)
r
where u is the velocity, ρ the fluid density assigned as ρ = ρ 1 f + ρ 2 (1 − f ) . The above conservation equations are
solved using the projection method and the VOF advection equation is solved with a PLIC (piecewise linear calculation)
algorithm [6]. Next, we present how we can apply the surface tension with the CSF and GFM method. To simplify the
presentation, we consider a static case with zero velocity.
Continuum Surface Tension Force (CSF) [1], [2]
In the CSF method, the surface tension force is represented as a continuum force per unit volume in regions along and
r
near to the interface. This force FS = σκ∇f appears as a source term in the momentum equation. Recently a consistent
formulation [2] was proposed in which the surface tension force appears in the right hand side of the pressure equation
and gradients are discretized at faces:
 1

 σκ (∇f ) face 
.
∇.
(∇P ) face  = ∇.
ρ

ρ
face
face




(4)
Ghost Fluid Methods (GFM) [3], [4]
In GFM, the pressure jump condition is applied as rigorously on the interface as the distance function φ allows. The
distance function is zero on the interface, negative inside the interface (fluid 1) and positive outside the interface (fluid
2). Since we are in the context of the VOF method, where we only store volume fractions, we need to reconstruct a
distance function. For that we have used our reconstruction distance function (RDF) technique [5]. First, normal
distances from the piecewise linear segments are computed using simple geometrical relations to all nearby cells. The
piecewise linear segments are constructed based on volume fractions [6] and are part of the VOF technique. The normal
distances to the nearby interfacial cells are then weighted to obtain the distance function. The weights are function of
the angle made by the interfacial normal and the vector between the cell center and the linear segment centroid (the
Mechanics of 21st Century - ICTAM04 Proceedings
smaller the angle, the greater the contribution). In GFM, the pressure stencil is modified to apply the jump condition at
the interface due to surface tension. The modifications appear in the right hand side of the pressure equation:
 1

∇.
(∇P ) face  = F L + F R + F T + F B ,
ρ
 face

(5)
where F L, FR, FT and F B are source terms. Nonzero terms are only those at faces across which the distance function
change signs. L is for left stencil (i-1,i), R for right stencil (i,i+1), T for top stencil (j,j+1) and B for bottom (j-1,j). For
example, the left stencil, if φ i , j ≤ 0 and φ i −1, j > 0 , becomes:
FL =
(1/ ρ )(i −1/ 2, j ) (σκ )I
( ∆x )2
(6)
Note that GFM require a series of tedious logical tests, which are not required in CSF method.
Curvature Estimation
To estimate interfacial curvature, a height function method is used, which has been found in [5] to be more accurate than
techniques using a distance function or volume fraction convolution.
RESULTS
To illustrate the difference between CSF and GFM we present here the results of a static computation, with dynamic
results forthcoming in the full paper. We consider a 2D drop of radius 0.25 in a unit square domain. The surface tension
is 1 and the curvature is imposed to its exact value 4 for not introducing error coming from its computation. The pressure
equations are solved using an iterative method Line SOR. Both methods recover the exact jump in pressure ∆P=4.
However, with the CSF method, the profile is smoother than the GFM result (see Figure 1). GFM are more accurate
because it takes account the exact interface position. Interestingly, both GFM and CSF results are independent of the
density ratio, as can easily be shown in their formulation.
Figure 1: Pressure profile using VOF-GFM and VOF-CSF along the x direction at y=0.5. Grid is resolution 20x20.
CONCLUSIONS
We have investigated the difference between the CSF and GFM to model the jump in pressure due to surface tension at
an interface. The GFM achieves a sharp jump in pressure, whereas the CSF method achieves a smooth transition. We
have demonstrated that we can combine the VOF and GFM by reconstructing distance functions from volume fractions
without solving an equation for the evolution of the distance function as in level-set techniques providing an alternative
to the CSF method. Further scrutiny of these two methods will be made with dynamical simulations.
ACKNOWLEDGEMENTS
This work is supported by the Advanced Simulation and Computing (ASC) Program in the U.S. Department of Energy.
References
[1] Brackbill J.U., Kothe D.B., Zemach C., A Continuum Method for Modelling Surface Tension, J. Comput. Phys., 100 :
335-354, 1992.
[2] Francois M.M., Kothe D.B., Dendy E.D., Sicilian J.M., Williams M.W., Balanced Force Implementation of the
Continuum Surface Tension Force Method into a Pressure Correction Algorithm, Proc. of the FEDSM’03, 4 th ASMEJSME Joint Fluids Engineering Conference, 2003.
[3] Liu X.-D., Fedkiw R.P., Kang M., A Boundary Condition Capturing Method for Poisson’s Equation on Irregular
Domains, J. Comput. Phys., 160 : 151-178, 2000.
[4] Kang M., Fedkiw R.P., Liu X.-D., A Boundary Condition Capturing Method for Multiphase Incompressible Flow, J.
Sci. Comput., 15:, 323-360, 2000.
[5] Cummins S.J., Francois M.M., Kothe D.B., Reconstructing Distance Functions from Volume Fractions: A Better Way
to Estimate Interface Topology ?, accepted for publication in Computers and Structures, LA -UR-03-7234, 2003.
[6] Rider W.J., Kothe D.B., Reconstructing Volume Tracking, J. Comput. Phys., 141 : 112-152, 1998.
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